# Fermi-Dirac thermal measurements: A framework for quantum hypothesis testing and semidefinite optimization

arXiv: https://arxiv.org/abs/2603.04061 · HTML: https://mlsanity.com/paper/2603.04061/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 82 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 14 of 14)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Quantum SDP-Solvers: Better upper and lower bounds](https://mlsanity.com/paper/1705.01843/) (2017)
- [Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models](https://mlsanity.com/paper/1609.02542/) (2016)
- [Quantum Boltzmann machine learning of ground-state energies](https://mlsanity.com/paper/2410.12935/) (2024)
- [Natural gradient and parameter estimation for quantum Boltzmann machines](https://mlsanity.com/paper/2410.24058/) (2024)
- [Variational Quantum Algorithms for Semidefinite Programming](https://mlsanity.com/paper/2112.08859/) (2021)
- [Quantum thermodynamics and semi-definite optimization](https://mlsanity.com/paper/2505.04514/) (2025)
- [Fast randomized entropically regularized semidefinite programming](https://mlsanity.com/paper/2303.12133/) (2023)
- [Quantum Fisher information matrices from R'enyi relative entropies](https://mlsanity.com/paper/2510.02218/) (2025)
- [Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems](https://mlsanity.com/paper/2508.09103/) (2025)
- [Five Starter Pieces: Quantum Information Science via Semi-definite Programs](https://mlsanity.com/paper/2112.08276/) (2021)
- [Digital Quantum Simulations of the Non-Resonant Open Tavis-Cummings Model](https://mlsanity.com/paper/2501.18522/) (2025)
- [Quantum Alternating Direction Method of Multipliers for Semidefinite Programming](https://mlsanity.com/paper/2510.10056/) (2025)

## Built on by (most cited first, 5 of 5)

- [Fermi-Dirac machines as quantizations of neurons](https://mlsanity.com/paper/2605.24386/) (2026)
- [Quantum principal component analysis without eigenvector recovery](https://mlsanity.com/paper/2605.27942/) (2026)
- [Canonical quantization of neurons](https://mlsanity.com/paper/2607.05000/) (2026)
- [Deep Learning for Semen Analysis in Male Infertility: Computer Vision, Multimodal Fusion, and Clinical Translation](https://mlsanity.com/paper/2607.05311/) (2026)
- [Unifying quantum measurement constructions via a relative-entropy minimum change principle](https://mlsanity.com/paper/2608.04055/) (2026)

## Abstract (arXiv)

Quantum measurements are the means by which we recover messages encoded into quantum states. They are at the forefront of quantum hypothesis testing, wherein the goal is to perform an optimal measurement for arriving at a correct conclusion. Mathematically, a measurement operator is Hermitian with eigenvalues in [0,1]. By noticing that this constraint on each eigenvalue is the same as that imposed on fermions by the Pauli exclusion principle, we interpret every eigenmode of a measurement operator as an independent effective fermionic mode. Under this perspective, various objective functions in quantum hypothesis testing can be viewed as the total expected energy associated with these fermionic occupation numbers. By instead fixing a temperature and minimizing the total expected fermionic free energy, we find that optimal measurements for these modified objective functions are Fermi-Dirac thermal measurements, wherein their eigenvalues are specified by Fermi-Dirac distributions. In the low-temperature limit, their performance closely approximates that of optimal measurements for quantum hypothesis testing, and we show that their parameters can be learned by classical or hybrid quantum-classical optimization algorithms. This leads to a new quantum machine-learning model, termed Fermi-Dirac machines, consisting of parameterized Fermi-Dirac thermal measurements-an alternative to quantum Boltzmann machines based on thermal states. Beyond hypothesis testing, we show how general semidefinite optimization problems can be solved using this approach, leading to a novel paradigm for semidefinite optimization on quantum computers, in which the goal is to implement thermal measurements rather than prepare thermal states. Finally, we propose quantum algorithms for implementing Fermi-Dirac thermal measurements, and we also propose second-order hybrid quantum-classical optimization algorithms.